For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

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1answer
73 views

Is there a (hypercomplex) number system, in which addition is **not** commutative

Now, we all know and love the quaternions, in which multiplication is not commutative, and the fact that, in all Clifford algebras, exponentiation is not commutative. Having looked at the properties ...
0
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0answers
33 views

Is there a term for extending a finite magma by adding coefficients from fields?

For example, the Quaternion numbers at their base have the Cayley table: $ * = \begin{bmatrix} 1 & i & j & k \\ i & -1 & k & -j \\ j & -k & -1 & i \\ k & j ...
3
votes
1answer
82 views

What is the meaning of quaternion interpolation?

Suppose I take the average between two quaternions, how does one see the meaning of the resulting rotation to make sure it is sensible, unlike interpolating Euler angles? I'm looking for an argument ...
0
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2answers
42 views

Rotation between two vectors as a function of time (one parameter rational motion design)

Given a time varying vector: $\mathbf{w}(t) = \mathbf{u} + t\mathbf{v}$ I would like to find a rotation matrix $\mathbf{R}(t)$ that rotates the positive x-axis $[1, 0, 0]^T$ onto the vector ...
8
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1answer
255 views

Riemann Zeta function, quaternions and physics

Disclaimer: This question is rather vague, and thus probably not suitable for Mathoverflow, so I prefer to ask it here. I'm sorry if it doesn't meet the standards of this site. Several years ago, I ...
3
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0answers
281 views

Tensor product of the spaces of quaternions and complex numbers

Let $ \mathbb{H} $ be the ring of quaternions and make the vector space $A = \mathbb{H} \otimes \mathbb{C}$ into a ring by defining $$(a \otimes w)(b \otimes z) = (ab \otimes wz) $$ for $a,b ...
4
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2answers
541 views

What is a good geometric interpretation of quaternion multiplication?

I understand that the formula for quaternion multiplication of $q_1=(s_1,\vec{v_1})$ by $q_2=(s_2,\vec{v_2})$ $q_1q_2=(s_1s_2-\vec{v_1}\cdot\vec{v_2}, \vec{v_1} \times\vec{v_2} + \vec{v_1}s_2 + ...
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0answers
90 views

How to obtain relative rotation?

I have two rotations, each of which can be described as a roll, pitch, and yaw (in radians): $$ r_1 = (3.14159, 1.57080, 1.6) $$ $$ r_2 = (3.14159, 1.57080, 1.4) $$ I am interested in the relative ...
4
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1answer
59 views

Proof that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$

In the wikipedia atrticle (http://en.wikipedia.org/wiki/Octonion) it is stated that "one can show that the subalgebra generated by any two elements of $\mathbb{O}$ is isomorphic to $\mathbb{R}$, ...
6
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4answers
157 views

What's the intuition for extending $\mathbb{C}$ to $\mathbb{H}$?

It seems to me that there is a clear, intuitive reason for extending the real number system to the complex number system. Namely, some polynomial equations that have no solutions in $\mathbb{R}$ ...
2
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0answers
40 views

Inner product space or Hilbert space of Quaternionic Functions

In what ways can you define an inner product, $<f,g>$, to create an inner product space or Hilbert space on the set of quaternionic functions $f:\mathbb{H} \rightarrow \mathbb{H}$?
6
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1answer
133 views

Do there exist equations that cannot be solved in $\mathbb{C}$, but can be solved in $\mathbb{H}$?

Excluding polynomials (whose solutions are covered by the Fundamental Theorem of Algebra), do there exist any univariable equations that cannot be solved in the complex numbers, but can be solved ...
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0answers
80 views

Prove that $Q$ is a group under quaternion multiplication

Consider the subset $Q$ of the quaternions defined by $$Q=\{1,-1,i,-i,j,-j,k,-k\}.$$ Show that $Q$ is a group under quaternion multiplication. I know to prove something's a group, you must show ...
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0answers
284 views

Transform a vector to global frame and ignore rotation about one axis or Full tilt compensated magnetometer

Good day everyone. I would like to lock the rotation about one specified axis. For example, let`s imagine that we have a quaternion which desribes the orientation of our rigid body relative to the ...
1
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1answer
44 views

Question about quaternionic conjugation

The quaternionic conjugation is defined by $$\begin{aligned}i &\mapsto -i\\j&\mapsto -j\\k&\mapsto -k\end{aligned}$$ But since $ij=k$, shouldn't we have that $k = ij \mapsto (-i)(-j) = ...
0
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1answer
66 views

Is there infinitely many “complex units”

As we know, $i$ = $\sqrt{-1}$, a simple complex unit. In complex space of two dimensions, you graph an axis of $a+bi$ where $i$ is your second dimension axis. Now, you also know, in three and four ...
3
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1answer
127 views

Quaternions, Lie Groups and Lie Algebras. Steps to realize a paper. [closed]

I have to realize a paper about quaternions and Lie Groups and Lie Algebras. How can I realize the links between quaternions and Lie Groups & Algebras. Which books do you recommend me? First, I ...
1
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1answer
95 views

Can different quaternions represent the same orientation?

For a current project I'm working on I have to use quaternions to represent the orientation of an object. The piece of code I'm working on now integrates rotation rates to the quaternion representing ...
4
votes
1answer
88 views

Are roots of unity in hypercomplex algebras well defined?

While playing around with cyclotomic fields, I started to wonder about taking the roots of unity in higher dimensional analogues of the complex plane. Are the roots of unity well defined in the ...
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2answers
225 views

Dual quaternion inverse

Is it true that for every dual quaternion $Q$ I can find it's inverse such that $QQ^{-1} = 1?$ Using the usual definition $Q^{-1}=\frac{Q^{*}}{||Q||^2}$ doesn't work for me, since the dual part ...
2
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0answers
47 views

Does Shor's algorithm work for noncommutitive or nonassociative algebras?

Shor's algorithm is said to solve the hidden subgroup problem for finite Abelian groups, at least according to the Wikipedia article I read. This means that it can be used to solve factoring or ...
0
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1answer
67 views

Motivation for construction of cross-product (Quaternions?)

I just found a very interesting article here: http://www.johndcook.com/blog/2012/02/15/dot-cross-and-quaternion-products/ The author observes that by defining i,j,k s.t. $i^2=j^2=k^2=ijk=-1$, ...
1
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1answer
149 views

Complex number with 3 dimensions [duplicate]

I was looking back on complex analysis and asked myself: ''Why is there no complex number in 3 dimensions ?''. To place this question let me define with what I mean with 3 dimensions in the following. ...
1
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1answer
36 views

Rotation orientations in n-dimensions

I'm doing a change of variables that involves doing simple rotations on the standard basis vectors in R^n, and I'm wondering what the standard orientations are in n dimensions are and why. For ...
2
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1answer
107 views

Is complex symplectic structure equivalent to a quaternionic structure?

Let $S$ be a vector space over $\mathbb{C}$ of dimension $\operatorname{dim} S =2n$, let me denote by $S_\mathbb{R}$ real form of $S$ i.e. vector space over $\mathbb{R}$ of dimension $4n$. Is it true ...
0
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1answer
202 views

How to rotate a 3d vector to be parallel to another 3d vector using quaternions?

I have a vector (a,b,c) and another vector (d,e,f). I'm trying to rotate (a,b,c) so its parallel to (d,e,f) using quaternions. I need help understanding how I would do this. I have so far that a ...
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2answers
57 views

Analogy between the quaternion ring and extensions of the rationals

I've started studing fields and their extensions. As an exercise I proved that $[\mathbb{Q}(\sqrt2,\sqrt3):\mathbb{Q}]=4$ by showing that $B=(1,\sqrt2,\sqrt3,\sqrt6)$ is a base for the extension field ...
0
votes
1answer
384 views

How can I transform coordinate systems with quaternions?

I have a coordinate system 0 which I'd first like to rotate about its z-Axis which gives me system 1, and afterwards rotate system 1 about its y-axis which gives me system 2. See picture: Both ...
3
votes
1answer
117 views

How to deal with multiple representations of quaternions

I'm using a quaternion to represent the orientation in a kalman filter. My algorithm works fine until I rotate "upside down". I think this is because there seems to be multiple ways to represent the ...
5
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2answers
154 views

Sources on Hamilton's Discovery of Quaternions

This is a strange question and I'm not sure where to put it; I'm currently writing an essay for a history of maths course, and I've chosen the topic of Hamilton's discovery of the quaternions. I ...
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1answer
161 views

Quaternion and Matrix

I have a quaternion for rotation and a matrix for changing axis(change coordinate from camera to my rendering scene ). I have tested two method and i except to have equal resuls but results are ...
0
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1answer
38 views

geometry rotation quaternion

Express the rotation of $\mathbb R^3$ by $\frac{\pi}{4}$ about the $x = y,\ z = 0$ axis by using quaternions and identifying $\mathbb R^3$ with $(i, j, k)$-space. Find the image of the point ...
14
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1answer
320 views

How to raise a number to a quaternion power

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: ...
5
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4answers
122 views

Show that $Q_8$ can't be embedded in $M_{2 \times 2}(\mathbb{R})$ as a group.

So, suppose that we're working in a field $F$. Consider the ring $M_{n \times n} (F)$ which is the set all $n \times n$ matrices with entries in $F$. Is it possible to determine whether a matrix ...
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0answers
70 views

Integral elements with predescribed properties in quaternion orders

In the course of doing some calculations I have found myself wanting to answer the following question: Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let ...
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0answers
52 views

Plücker coordinates of the Clifford parallels

Let $$q=\cos\theta+(x_q\textbf{i}+y_q\textbf{j}+z_q\textbf{k})\sin\theta$$ be a unit quaternion parameterised by $\theta\in\mathbb{R}$, where $(x_q,y_q,z_q)$ is fixed and $x_q^2+y_q^2+z_q^2=1$, and ...
0
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1answer
89 views

Quaternion isn't normal!

I use the following scheme: Quaternion: [ w, x, y, z ] Then I use Euler Angles and convert it to a quaternion, as following: ...
0
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1answer
112 views

non-division quaternion algebra is isomorphic to $2\times 2$ matrices

Let $k$ be a field of characteristic $\neq2$. Let $a,b\in k$ be nonzero elements. Let $A:=\left(\frac{a,b}{k}\right)$ be the quaternion algebra over $k$ with parameters $a,b$. Suppose $A$ is not a ...
3
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2answers
4k views

Getting Euler (Tait-Bryan) Angles from Quaternion representation

Apologies if this has already been answered, but I haven't been able to get a clear answer from looking on Stack Exchange so-far. I'm trying to solve a camera stabilization problem. I have a 2-axis ...
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1answer
76 views

How can I find a unit velocity vector between two quaternions?

I have two quaternions, $Q_0$ and $Q_1$. I want to find the unit angular velocity vector $w$ that rotates $Q_0$ in the direction of $Q_1$ (shortest path). How can I do this? The analog of what I ...
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1answer
235 views

Quaternion derivative w.r.t. its angle

The following quaternion represents a rotation by $\theta$ around the z-axis: \begin{align} q &= (\cos(\frac{1}{2}\theta), \vec{u}\cdot\sin(\frac{1}{2}\theta)), \\ \vec{u}&=(0,0,1)^t ...
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0answers
123 views

Indefinite quaternion algebra over Q

Let $D$ be an indefinite quaternion algebra over $\Bbb Q$. We have a chosen isomorphism $\iota \colon D \otimes_{\Bbb Q} {\Bbb R} \cong {\mathrm{M}}_2({\Bbb R})$. Q: If we choose another isomorphism ...
4
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1answer
63 views

Preimage of a point by a power map in quaternions

Suppose we have a point $x_0\in{\bf H}$ (where by $\bf H$ I denote the ring of quaternions). What I'm curious about is what can the set of solutions of $x^2=x_0$ look like? From what I've checked, ...
0
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1answer
535 views

Quaternion Decomposition

I'm having trouble decomposing a unit quaternion into euler angles (or roll, pitch and yaw). The overall goal is to tell how a phone is rotated with respect to the world. I'm given a unit quaternion ...
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1answer
148 views

About the derivation of a composite quaternion

This problem has been bothering me for several days, hence I decided to ask you for help. I am reading the book "Quaternions and Rotation Sequence" written by Jack B. Kuipers. In section 6.4, the ...
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0answers
83 views

Dual quaternion derivation

I'd like to derivate a dual quaternion \begin{align} \hat{q}&=(1 + \frac{1}{2}\epsilon\vec{t})q \end{align} where \begin{align} q &= e^\vec{w} , \\\vec{w}&=(0, w_1,w_2,w_3)^t ...
1
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1answer
126 views

Unit quaternions as rotations

How would one represent the map $f$ such that $f(1) = i, f(i) = -1$ and keeping $j$ and $k$ fixed as a quaternion representation of rotations?
2
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1answer
115 views

Quaternions and critically damped spring

I would like to apply critically damped spring smoothing method to smooth movement on the unit sphere to a desired orientation. I have two quaternions, one that represent current orientation and one ...
0
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2answers
165 views

How to find angle difference in quatenions?

How does one find the angle difference between two quaternions. There was an answer to this post which says the angle difference between $x$ and $y$ is $z=x\ast \mathrm{conj}(y)$. Is that the ...
0
votes
1answer
99 views

smooth orientation change with quaternions

My camera orientation is looking in the $v_1$ direction. Something happens on direction $v_2$ and I want the camera to move smoothly to look at that direction. So, to find the quaternion to go from ...